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The domain of csc x was given to be all x such that x k π for any integer k . Would the domain of y = A csc ( B x C ) + D be x C + k π B ?

Yes. The excluded points of the domain follow the vertical asymptotes. Their locations show the horizontal shift and compression or expansion implied by the transformation to the original function’s input.

Given a function of the form y = A csc ( B x ) , graph one period.

  1. Express the function given in the form y = A csc ( B x ) .
  2. | A | .
  3. Identify B and determine the period, P = 2 π | B | .
  4. Draw the graph of y = A sin ( B x ) .
  5. Use the reciprocal relationship between y = sin x and y = csc x to draw the graph of y = A csc ( B x ) .
  6. Sketch the asymptotes.
  7. Plot any two reference points and draw the graph through these points.

Graphing a variation of the cosecant function

Graph one period of f ( x ) = −3 csc ( 4 x ) .

  • Step 1. The given function is already written in the general form, y = A csc ( B x ) .
  • Step 2. | A | = | 3 | = 3 , so the stretching factor is 3.
  • Step 3. B = 4 , so P = 2 π 4 = π 2 . The period is π 2 units.
  • Step 4. Sketch the graph of the function g ( x ) = −3 sin ( 4 x ) .
  • Step 5. Use the reciprocal relationship of the sine and cosecant functions to draw the cosecant function .
  • Steps 6–7. Sketch three asymptotes at x = 0 , x = π 4 , and x = π 2 . We can use two reference points, the local maximum at ( π 8 , −3 ) and the local minimum at ( 3 π 8 , 3 ) . [link] shows the graph.
    A graph of one period of a cosecant function. There are vertical asymptotes at x=0, x=pi/4, and x=pi/2.
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Graph one period of f ( x ) = 0.5 csc ( 2 x ) .

A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.
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Given a function of the form f ( x ) = A csc ( B x C ) + D , graph one period.

  1. Express the function given in the form y = A csc ( B x C ) + D .
  2. Identify the stretching/compressing factor, | A | .
  3. Identify B and determine the period, 2 π | B | .
  4. Identify C and determine the phase shift, C B .
  5. Draw the graph of y = A csc ( B x ) but shift it to the right by and up by D .
  6. Sketch the vertical asymptotes, which occur at x = C B + π | B | k , where k is an integer.

Graphing a vertically stretched, horizontally compressed, and vertically shifted cosecant

Sketch a graph of y = 2 csc ( π 2 x ) + 1. What are the domain and range of this function?

  • Step 1. Express the function given in the form y = 2 csc ( π 2 x ) + 1.
  • Step 2. Identify the stretching/compressing factor, | A | = 2.
  • Step 3. The period is 2 π | B | = 2 π π 2 = 2 π 1 2 π = 4.
  • Step 4. The phase shift is 0 π 2 = 0.
  • Step 5. Draw the graph of y = A csc ( B x ) but shift it up D = 1.
  • Step 6. Sketch the vertical asymptotes, which occur at x = 0 , x = 2 , x = 4.

The graph for this function is shown in [link] .

A graph of 3 periods of a modified cosecant function, with 3 vertical asymptotes, and a dotted sinusoidal function that has local maximums where the cosecant function has local minimums and local minimums where the cosecant function has local maximums.
A transformed cosecant function
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Given the graph of f ( x ) = 2 cos ( π 2 x ) + 1 shown in [link] , sketch the graph of g ( x ) = 2 sec ( π 2 x ) + 1 on the same axes.

A graph of two periods of a modified cosine function. Range is [-1,3], graphed from x=-4 to x=4.
A graph of two periods of both a secant and consine function. Grpah shows that cosine function has local maximums where secant function has local minimums and vice versa.
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Analyzing the graph of y = cot x

The last trigonometric function we need to explore is cotangent    . The cotangent is defined by the reciprocal identity cot x = 1 tan x . Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at 0 , π , etc. Since the output of the tangent function is all real numbers, the output of the cotangent function is also all real numbers.

We can graph y = cot x by observing the graph of the tangent function because these two functions are reciprocals of one another. See [link] . Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.

Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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